Question

The percent of fat calories that a person consumes each day is normally distributed with a mean of 35 and a standard deviation of 10. Suppose that 25 individuals are randomly chosen. Let X = average percent of fat calories.

(a) For the group of 25, find the probability that the average percent of fat calories consumed is more than thirty-seven. (Round your answer to four decimal places)

b) Find the first quartile for the average percent of fat calories. (Round your answer to two decimal places.)
percent of fat calories

Answer 1

that ,

mean = $\mu$ = 35

standard deviation = $\sigma$ = 10

n = 25

$\mu$_{$\bar x$} = 35

$\sigma$_{$\bar x$} = $\sigma$ / $\sqrt$ n = 10/ $\sqrt$ 25= 2

P($\bar x$ > 37) = 1 - P($\bar x$ < 37)

= 1 - P[($\bar x$ - $\mu$ _{$\bar x$} ) / $\sigma$ _{$\bar x$} < (37-35 /2 ]

= 1 - P(z < 1)

Using z table,    

= 1 - 0.8413

= 0.1587

b.

Using standard normal table,

P(Z < z) = 25%

= P(Z < z) = 0.25  

= P(Z < ) = 0. 25

z = -0.67

Using z-score formula  

x= z * $\sigma$ + $\mu$

x= -0.67 *10+35

x=28.3